Kepler’s laws of planetary motion

In astronomy, Kepler’s laws of planetary motion are three scientific laws describing the motion of planets around the Sun. Kepler’s laws are now traditionally enumerated in this way:

 

Figure 1: Illustration of Kepler’s three laws with two planetary orbits.
(1) The orbits are ellipses, with focal points ƒ1 and ƒ2for the first planet and ƒ1 and ƒ3 for the second planet. The Sun is placed in focal point ƒ1.

(2) The two shaded sectors A1 and A2 have the same surface area and the time for planet 1 to cover segment A1 is equal to the time to cover segment A2.

(3) The total orbit times for planet 1 and planet 2 have a ratio a13/2 : a23/2.

  1. The orbit of a planet is an ellipse with the Sun at one of the two foci.
  2. A line segment joining a planet and the Sun sweeps out equal areas during equal intervals of time.
  3. The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

Most planetary orbits are almost circles, so it is not apparent that they are actually ellipses. Calculations of the orbit of the planet Mars first indicated to Kepler its elliptical shape, and he inferred that other heavenly bodies, including those farther away from the Sun, have elliptical orbits also. Kepler’s work broadly followed the heliocentric theory of Nicolaus Copernicusby asserting that the Earth orbited the Sun. It innovated in explaining how the planets’ speeds varied, and using elliptical orbits rather than circular orbits with epicycles.

Isaac Newton showed in 1687 that relationships like Kepler’s would apply in the solar system to a good approximation, as consequences of his own laws of motion and law of universal gravitation. Together with Newton’s theories, Kepler’s laws became part of the foundation of modern astronomy and physics.

Nomenclature

It took nearly two centuries for the current formulation of Kepler’s work to take on its settled form. Voltaire’s Eléments de la philosophie de Newton (Elements of Newton’s Philosophy) of 1738 was the first publication to use the terminology of “laws”. The Biographical Encyclopedia of Astronomersin its article on Kepler (p. 620) states that the terminology of scientific laws for these discoveries was current at least from the time of Joseph de Lalande. It was the exposition of Robert Small, in An account of the astronomical discoveries of Kepler (1804) that made up the set of three laws, by adding in the third. Small also claimed, against the history, that these were empirical laws, based on inductive reasoning.

Further, the current usage of “Kepler’s Second Law” is something of a misnomer. Kepler had two versions of it, related in a qualitative sense, the “distance law” and the “area law”. The “area law” is what became the Second Law in the set of three; but Kepler did himself not privilege it in that way.

History

Johannes Kepler published his first two laws about planetary motion in 1609, having found them by analyzing the astronomical observations of Tycho Brahe. Kepler’s third law was published in 1619.

Kepler in 1621 and Godefroy Wendelin in 1643 noted that Kepler’s third law applies to the four brightest moons of Jupiter. The second law (“area law” form) was contested by Nicolaus Mercator in a book from 1664; but by 1670 he was publishing in its favour in Philosophical Transactions, and as the century proceeded it became more widely accepted. The reception in Germany changed noticeably between 1688, the year in which Newton’s Principia was published and was taken to be basically Copernican, and 1690, by which time work of Gottfried Leibniz on Kepler had been published.

Newton is credited with understanding that the second law is not special to the inverse square law of gravitation, being a consequence just of the radial nature of that law; while the other laws do depend on the inverse square form of the attraction. Carl Runge and Wilhelm Lenz much later identified a symmetry principle in the phase space of planetary motion (the orthogonal group O(4) acting) which accounts for the first and third laws in the case of Newtonian gravitation, as conservation of angular momentum does via rotational symmetry for the second law.

First law

The orbit of every planet is an ellipse with the Sun at one of the two foci.

Figure 2: Kepler’s first law placing the Sun at the focus of an elliptical orbit

Figure 4: Heliocentric coordinate system (r, θ) for ellipse. Also shown are: semi-major axis a, semi-minor axisb and semi-latus rectum p; center of ellipse and its two foci marked by large dots. For θ = 0°, r = rmin and for θ = 180°, r = rmax.

Mathematically, an ellipse can be represented by the formula:

r=frac{p}{1+varepsilon, costheta},

where p is the semi-latus rectum, and ε is the eccentricity of the ellipse, and r is the distance from the Sun to the planet, and θ is the angle to the planet’s current position from its closest approach, as seen from the Sun. So (rθ) are polar coordinates.

For an ellipse 0 < ε < 1 ; in the limiting case ε = 0, the orbit is a circle with the sun at the centre (see section Zero eccentricity below).

Second law

A line joining a planet and the Sun sweeps out equal areas during equal intervals of time.[1]

The same blue area is swept out in a given time. The green arrow is velocity. The purple arrow directed towards the Sun is the acceleration. The other two purple arrows are acceleration components parallel and perpendicular to the velocity

The orbital radius and angular velocity of the planet in the elliptical orbit will vary. This is shown in the animation where the planet travels faster when close to the sun, then slower when far from the sun. Kepler’s second law states that the blue sector has constant area.

Third law

The square of the orbital period of a planet is directly proportional to the cube of the semi-major axis of its orbit.

This captures the relationship between the distance of planets from the Sun, and their orbital periods.

For a brief biography of Kepler and discussion of his third law, see: NASA: Stargaze

Kepler enunciated in 1619 this third law in a laborious attempt to determine what he viewed as the “music of the spheres” according to precise laws, and express it in terms of musical notation.[16] So it was known as the harmonic law. [17]

Mathematically, the law says that the expression

 P^2/a^3

has the same value for all the planets in the solar system. Here P is the time taken for a planet to complete an orbit round the sun, anda is the mean value between the maximum and minimum distances between the planet and sun.

An alternative method for calculation of the third law has been described by William R. Livingston in a paper titled “A Cube Root Method for Recalculation of Kepler’s Third Law”.

Formulary

The mathematical model of the kinematics of a planet subject to the laws allows a large range of further calculations.

First law and the geometry of the ellipse

At θ = 0°, perihelion, the distance is minimum

r_mathrm{min}=frac{p}{1+varepsilon}.

At θ = 90° and at θ = 270°, the distance is equal to the semi-latus rectum.

At θ = 180°, aphelion, the distance is maximum

r_mathrm{max}=frac{p}{1-varepsilon}.

The semi-major axis l is the arithmetic mean between rmin and rmax:

,r_max - a=a-r_min
a=frac{p}{1-varepsilon^2}.

The semi-minor axis b is the geometric mean between rmin and rmax:

frac{r_max} b =frac b{r_min}
b=frac p{sqrt{1-varepsilon^2}}.

The semi-latus rectum p is the harmonic mean between rmin and rmax:

frac{1}{r_min}-frac{1}{p}=frac{1}{p}-frac{1}{r_max}
pa=r_max r_min=b^2,.

The eccentricity ε is the coefficient of variation between rmin and rmax:

varepsilon=frac{r_mathrm{max}-r_mathrm{min}}{r_mathrm{max}+r_mathrm{min}}.

The area of the ellipse is

A=pi a b,.

The special case of a circle is ε = 0, resulting in r = p = rmin = rmax = a = b and A = π r2.

Second law, mathematical derivation

In a small time dt, the planet sweeps out a small triangle having base line r, and height r dtheta, and area dA=tfrac 1 2cdot rcdot r dtheta and so the constant areal velocity is frac{dA}{dt}=tfrac{1}{2}r^2 frac{dtheta}{dt}.

The area enclosed by the elliptical orbit is pi ab., So the period P, satisfies

Pcdot tfrac 12r^2 frac{dtheta}{dt}=pi a b

and the mean motion of the planet around the Sun

n = {2pi}/P

satisfies

r^2{dtheta} = a b n dt .

Third law, modern formulation

The modern formulation, with the constant evaluated, reads as:

frac{T^2}{r^3} = frac{4 pi^2}{GM}

where

  • T is the orbital period of the orbiting body,
  • M is the mass of the star,
  • G is the universal gravitational constant and
  • r is the radius, i.e. the semi-major axis of the ellipse.

In the full formulation under Newton’s laws of motion, M should be replaced by

M+m,

where m is the mass of the orbiting body. Consequently, the proportionality constant is not truly the same for each planet. Nevertheless, given that m is so small relative to Mfor planets in our solar system, the approximation is good in the original setting.

Zero eccentricity

Kepler’s laws refine the model of Copernicus, which assumed circular orbits. If the eccentricity of a planetary orbit is zero, then Kepler’s laws state:

  1. The planetary orbit is a circle with the Sun at the center
  2. The speed of the planet in the orbit is constant
  3. The square of the sidereal period is proportionate to the cube of the distance from the Sun.

Actually, the eccentricities of the orbits of the six planets known to Copernicus and Kepler are quite small, so the rules above give excellent approximations of planetary motion, but Kepler’s laws fit observations even better.

Kepler’s corrections to the Copernican model are not at all obvious:

  1. The planetary orbit is not a circle, but an ellipse
  2. The Sun is not at the center but at a focal point
  3. Neither the linear speed nor the angular speed of the planet in the orbit is constant, but the area speed is constant.
  4. The square of the sidereal period is proportionate to the cube of the mean between the maximum and minimum distances from the Sun.

The nonzero eccentricity of the orbit of the earth makes the time from the March equinox to the September equinox, around 186 days, unequal to the time from the September equinox to the March equinox, around 179 days. A diameter would cut the orbit into equal parts, but the plane through the sun parallel to the equator of the earth cuts the orbit into two parts with areas in a 186 to 179 ratio, so the eccentricity of the orbit of the Earth is approximately

varepsilonapproxfrac pi 4 frac {186-179}{186+179}approx 0.015,

which is close to the correct value (0.016710219). (See Earth’s orbit). The calculation is correct when the perihelion, the date that the Earth is closest to the Sun, is on asolstice. The current perihelion, near January 4, is fairly close to the solstice on December 21 or 22.

Planetary acceleration

A sudden sunward velocity change is applied to a planet. Then the areas of the triangles defined by the path of the planet for fixed time intervals will be equal. (Click on image for a detailed description.)

Isaac Newton computed in his Philosophiæ Naturalis Principia Mathematica the acceleration of a planet moving according to Kepler’s first and second law.

  1. The direction of the acceleration is towards the Sun.
  2. The magnitude of the acceleration is in inverse proportion to the square of the distance from the Sun.

This suggests that the Sun may be the physical cause of the acceleration of planets.

Newton defined the force on a planet to be the product of its mass and the acceleration. (See Newton’s laws of motion). So:

  1. Every planet is attracted towards the Sun.
  2. The force on a planet is in direct proportion to the mass of the planet and in inverse proportion to the square of the distance from the Sun.

Here the Sun plays an unsymmetrical part, which is unjustified. So he assumed Newton’s law of universal gravitation:

  1. All bodies in the solar system attract one another.
  2. The force between two bodies is in direct proportion to the product of their masses and in inverse proportion to the square of the distance between them.

As the planets have small masses compared to that of the Sun, the orbits conform to Kepler’s laws approximately. Newton’s model improves upon Kepler’s model and fits actual observations more accurately. (See two-body problem).

A deviation in the motion of a planet from Kepler’s laws due to the gravity of other planets is called a perturbation.

Below comes the detailed calculation of the acceleration of a planet moving according to Kepler’s first and second laws.

Acceleration vector[edit]

See also: Polar coordinate § Vector calculus and Mechanics of planar particle motion

From the heliocentric point of view consider the vector to the planet mathbf{r} = r hat{mathbf{r}} where  r is the distance to the planet and the direction  hat {mathbf{r}} is a unit vector. When the planet moves the directions change:

 frac{dhat{mathbf{r}}}{dt}=dot{hat{mathbf{r}}} = dottheta hat{boldsymboltheta},qquad frac{dhat{boldsymboltheta}}{dt}=dot{hat{boldsymboltheta}} = -dottheta hat{mathbf{r}}

where hat{boldsymboltheta} is the unit vector orthogonal to hat{mathbf{r}} and pointing in the direction of rotation, and theta is the polar angle, and where a dot on top of the variable signifies differentiation with respect to time.

So differentiating the position vector twice to obtain the velocity and the acceleration vectors:

dot{mathbf{r}} =dot{r} hat{mathbf{r}} + r dot{hat{mathbf{r}}} =dot{r} hat{mathbf{r}} + r dot{theta} hat{boldsymbol{theta}},
ddot{mathbf{r}} = (ddot{r} hat{mathbf{r}} +dot{r} dot{hat{mathbf{r}}} ) + (dot{r}dot{theta} hat{boldsymbol{theta}} + rddot{theta} hat{boldsymbol{theta}} + rdot{theta} dot{hat{boldsymbol{theta}}}) = (ddot{r} - rdot{theta}^2) hat{mathbf{r}} + (rddot{theta} + 2dot{r} dot{theta}) hat{boldsymbol{theta}}.

So

ddot{mathbf{r}} = a_r hat{boldsymbol{r}}+a_thetahat{boldsymbol{theta}}

where the radial acceleration is

a_r=ddot{r} - rdot{theta}^2

and the transversal acceleration is

a_theta=rddot{theta} + 2dot{r} dot{theta}.

The inverse square law[edit]

Kepler’s laws say that

r^2dot theta = nab

is constant.

The transversal acceleration a_theta is zero:

frac{d (r^2 dot theta)}{dt} = r (2 dot r dot theta + r ddot theta ) = r a_theta = 0.

So the acceleration of a planet obeying Kepler’s laws is directed towards the sun.

The radial acceleration a_r is

a_r = ddot r - r dot theta^2= ddot r - r left(frac{nab}{r^2} right)^2= ddot r -frac{n^2a^2b^2}{r^3}.

Kepler’s first law states that the orbit is described by the equation:

frac{p}{r} = 1+ varepsilon costheta.

Differentiating with respect to time

-frac{pdot r}{r^2} = -varepsilon sin theta ,dot theta

or

pdot r = nab,varepsilonsin theta.

Differentiating once more

pddot r =nab varepsilon cos theta ,dot theta =nab varepsilon cos theta ,frac{nab}{r^2} =frac{n^2a^2b^2}{r^2}varepsilon cos theta .

The radial acceleration a_r satisfies

p a_r = frac{n^2 a^2b^2}{r^2}varepsilon cos theta - pfrac{n^2 a^2b^2}{r^3} = frac{n^2a^2b^2}{r^2}left(varepsilon cos theta - frac{p}{r}right).

Substituting the equation of the ellipse gives

p a_r = frac{n^2a^2b^2}{r^2}left(frac p r - 1 - frac p rright)= -frac{n^2a^2}{r^2}b^2.

The relation b^2=pa gives the simple final result

a_r=-frac{n^2a^3}{r^2}.

This means that the acceleration vector mathbf{ddot r} of any planet obeying Kepler’s first and second law satisfies the inverse square law

mathbf{ddot r} = - frac{alpha}{r^2}hat{mathbf{r}}

where

alpha = n^2 a^3,

is a constant, and hat{mathbf r} is the unit vector pointing from the Sun towards the planet, and r, is the distance between the planet and the Sun.

According to Kepler’s third law, alpha has the same value for all the planets. So the inverse square law for planetary accelerations applies throughout the entire solar system.

The inverse square law is a differential equation. The solutions to this differential equation include the Keplerian motions, as shown, but they also include motions where the orbit is a hyperbola or parabola or a straight line. See Kepler orbit.

Newton’s law of gravitation[edit]

By Newton’s second law, the gravitational force that acts on the planet is:

mathbf{F} = m_{Planet} mathbf{ddot r} = - {m_{Planet} alpha}{r^{-2}}hat{mathbf{r}}

where m_{Planet} is the mass of the planet and alpha has the same value for all planets in the solar system. According to Newton’s third Law, the Sun is attracted to the planet by a force of the same magnitude. Since the force is proportional to the mass of the planet, under the symmetric consideration, it should also be proportional to the mass of the Sun,m_{Sun}. So

alpha = Gm_{Sun}

where G is the gravitational constant.

The acceleration of solar system body number i is, according to Newton’s laws:

mathbf{ddot r_i} = Gsum_{jne i} {m_j}{r_{ij}^{-2}}hat{mathbf{r}}_{ij}

where m_j is the mass of body j, r_{ij} is the distance between body i and body j, hat{mathbf{r}}_{ij} is the unit vector from body i towards body j, and the vector summation is over all bodies in the world, besides i itself.

In the special case where there are only two bodies in the world, Earth and Sun, the acceleration becomes

mathbf{ddot r}_{Earth} = G{m_{Sun}}{r_{{Earth},{Sun}}^{-2}}hat{mathbf{r}}_{{Earth},{Sun}}

which is the acceleration of the Kepler motion. So this Earth moves around the Sun according to Kepler’s laws.

If the two bodies in the world are Moon and Earth the acceleration of the Moon becomes

mathbf{ddot r}_{Moon} = G{m_{Earth}}{r_{{Moon},{Earth}}^{-2}}hat{mathbf{r}}_{{Moon},{Earth}}

So in this approximation the Moon moves around the Earth according to Kepler’s laws.

In the three-body case the accelerations are

mathbf{ddot r}_{Sun} = G m_{Earth}{r_{{Sun},{Earth}}^{-2}}hat{mathbf{r}}_{{Sun},{Earth}} + G{m_{Moon}}{r_{{Sun},{Moon}}^{-2}}hat{mathbf{r}}_{{Sun},{Moon}}
mathbf{ddot r}_{Earth} = G{m_{Sun}}{r_{{Earth},{Sun}}^{-2}}hat{mathbf{r}}_{{Earth},{Sun}} + G{m_{Moon}}{r_{{Earth},{Moon}}^{-2}}hat{mathbf{r}}_{{Earth},{Moon}}
mathbf{ddot r}_{Moon} = G{m_{Sun}}{r_{{Moon},{Sun}}^{-2}}hat{mathbf{r}}_{{Moon},{Sun}}+G{m_{Earth}}{r_{{Moon},{Earth}}^{-2}}hat{mathbf{r}}_{{Moon},{Earth}}

These accelerations are not those of Kepler orbits, and the three-body problem is complicated. But Keplerian approximation is the basis for perturbation calculations. See Lunar theory.

Position as a function of time 

Kepler used his two first laws to compute the position of a planet as a function of time. His method involves the solution of a transcendental equation called Kepler’s equation.

The procedure for calculating the heliocentric polar coordinates (r,θ) of a planet as a function of the time t since perihelion, is the following four steps:

1. Compute the mean anomaly M=ncdot t where n is the mean motion.

ncdot P=2pi where P is the period.
2. Compute the eccentric anomaly E by solving Kepler’s equation:

 M=E-varepsiloncdotsin E
3. Compute the true anomaly θ by the equation:

(1-varepsilon)cdottan^2frac theta 2 = (1+varepsilon)cdottan^2frac E 2
4. Compute the heliocentric distance r from the first law:

rcdot(1+varepsiloncdotcostheta)=acdot(1-varepsilon^2)

The important special case of circular orbit, ε = 0, gives θ = E = M. Because the uniform circular motion was considered to be normal, a deviation from this motion was considered an anomaly.

The proof of this procedure is shown below.

Mean anomaly, M[edit]

FIgure 5: Geometric construction for Kepler’s calculation of θ. The Sun (located at the focus) is labeled S and the planet P. The auxiliary circle is an aid to calculation. Line xd is perpendicular to the base and through the planetP. The shaded sectors are arranged to have equal areas by positioning of point y.

The Keplerian problem assumes an elliptical orbit and the four points:

s the Sun (at one focus of ellipse);
z the perihelion
c the center of the ellipse
p the planet

and

 a=|cz|, distance between center and perihelion, the semimajor axis,
 varepsilon={|cs|over a}, the eccentricity,
 b=asqrt{1-varepsilon^2}, the semiminor axis,
 r=|sp| , the distance between Sun and planet.
theta=angle zsp, the direction to the planet as seen from the Sun, the true anomaly.

The problem is to compute the polar coordinates (r,θ) of the planet from the time since perihelion, t.

It is solved in steps. Kepler considered the circle with the major axis as a diameter, and

 x, the projection of the planet to the auxiliary circle
 y, the point on the circle such that the sector areas |zcy| and |zsx| are equal,
M=angle zcy, the mean anomaly.

The sector areas are related by |zsp|=frac b a cdot|zsx|.

The circular sector area  |zcy| = frac{a^2 M}2.

The area swept since perihelion,

|zsp|=frac b a cdot|zsx|=frac b a cdot|zcy|=frac b acdotfrac{a^2 M}2 = frac {a b M}{2},

is by Kepler’s second law proportional to time since perihelion. So the mean anomaly, M, is proportional to time since perihelion, t.

M=n t,

where n is the mean motion.

Eccentric anomaly, E

When the mean anomaly M is computed, the goal is to compute the true anomaly θ. The function θ=f(M) is, however, not elementary.[19] Kepler’s solution is to use

E=angle zcx, x as seen from the centre, the eccentric anomaly

as an intermediate variable, and first compute E as a function of M by solving Kepler’s equation below, and then compute the true anomaly θ from the eccentric anomaly E. Here are the details.

 |zcy|=|zsx|=|zcx|-|scx|
frac{a^2 M}2=frac{a^2 E}2-frac {avarepsiloncdot asin E}2

Division by a2/2 gives Kepler’s equation

M=E-varepsiloncdotsin E.

This equation gives M as a function of E. Determining E for a given M is the inverse problem. Iterative numerical algorithms are commonly used.

Having computed the eccentric anomaly E, the next step is to calculate the true anomaly θ.

True anomaly, θ[edit]

Note from the figure that

overrightarrow{cd}=overrightarrow{cs}+overrightarrow{sd}

so that

acdotcos E=acdotvarepsilon+rcdotcos theta.

Dividing by a and inserting from Kepler’s first law

 frac r a =frac{1-varepsilon^2}{1+varepsiloncdotcos theta}

to get

cos E =varepsilon+frac{1-varepsilon^2}{1+varepsiloncdotcos theta}cdotcos theta =frac{varepsiloncdot(1+varepsiloncdotcos theta)+(1-varepsilon^2)cdotcos theta}{1+varepsiloncdotcos theta} =frac{varepsilon +cos theta}{1+varepsiloncdotcos theta}.

The result is a usable relationship between the eccentric anomaly E and the true anomaly θ.

A computationally more convenient form follows by substituting into the trigonometric identity:

tan^2frac{x}{2}=frac{1-cos x}{1+cos x}.

Get

tan^2frac{E}{2} =frac{1-cos E}{1+cos E} =frac{1-frac{varepsilon+cos theta}{1+varepsiloncdotcos theta}}{1+frac{varepsilon+cos theta}{1+varepsiloncdotcos theta}} =frac{(1+varepsiloncdotcos theta)-(varepsilon+cos theta)}{(1+varepsiloncdotcos theta)+(varepsilon+cos theta)} =frac{1-varepsilon}{1+varepsilon}cdotfrac{1-cos theta}{1+cos theta}=frac{1-varepsilon}{1+varepsilon}cdottan^2frac{theta}{2}.

Multiplying by 1+ε gives the result

(1-varepsilon)cdottan^2frac theta 2 = (1+varepsilon)cdottan^2frac E 2

This is the third step in the connection between time and position in the orbit.

Distance, 

The fourth step is to compute the heliocentric distance r from the true anomaly θ by Kepler’s first law:

 rcdot(1+varepsiloncdotcos theta)=acdot(1-varepsilon^2)

Using the relation above between θ and E the final equation for the distance r is:

 r=acdot(1-varepsiloncdotcos E).

Source: Wikipedia

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